Two-Mode Bosonic State Tomography with Single-Shot Joint-Parity Measurement of a Trapped Ion

TL;DR

The paper presents a direct, single-shot joint-parity measurement scheme for multimode bosonic states in a trapped-ion system, enabling efficient Wigner-function tomography and model-based density-matrix estimation. By implementing a spin-dependent beam splitter via bichromatic Raman detuning, joint phonon parity is encoded into the spin phase and read out non-destructively, allowing tomography of two-mode Fock states and entangled coherent states. The authors quantify state fidelity and purity (fidelities around 0.66, purities near 0.46–0.48), demonstrate parity-based error mitigation through post-selection, and analyze error sources to assess scalability to additional modes. This approach advances continuous-variable quantum information processing in trapped ions and holds potential for metrology and CV quantum computing across platforms.

Abstract

The full characterization of a continuous-variable quantum system is a challenging problem. For the trapped-ion system, a number of methods of measuring the quantum states have been developed, including the measurement of the Q quasiprobability function and the density-matrix elements in the Fock basis, but these approaches are often slow and difficult to scale to multimode states. Here, we demonstrate a novel and powerful scheme for measuring a continuous-variable quantum state that uses the direct single-shot measurement of the joint parity of the phonon states of a trapped ion. We drive a spin-dependent bichromatic beam-splitter interaction that coherently exchanges phonons between different harmonic oscillator modes of the ion. This interaction encodes the joint-parity information into the relative phase between the two spin states, enabling measurement of the combined phonon-number parity across multiple modes in a single shot. Leveraging this capability, we directly measure multimode Wigner quasiprobability distributions to perform quantum state tomography of an entangled coherent state, and calculate various quantum informational quantities with a model-based estimation of the density matrix. We further show that the single-shot joint-parity measurement can be used to detect parity-flip errors in real time. By postselecting the parity-measurement outcomes, we experimentally demonstrate the partial recovery of coherence, effectively implementing an error-mitigation technique. Lastly, we identify the various sources of error affecting the fidelity of the spin-dependent beam-splitter operation and study the feasibility of high-fidelity operations. The interaction studied in this work can be extended to more than two modes, and is highly relevant to continuous-variable quantum computing and quantum metrology.

Figures (16)

• Figure 1: Experimental sequence for the two-mode Wigner function measurement. After initializing the spin to $\ket{\downarrow}$ and preparing the target state $\ket{\psi}_{12}=U_\psi\ket{0}_1\ket{0}_2$, coherent displacements---$\operatorname{SDF}_1(-\beta_1)$ and $\operatorname{SDF}_2(-\beta_2)$---are applied in the respective phase spaces to probe the Wigner function $W(\beta_1, \beta_2)$. The diamond-shaped control symbols represent the spin-dependence in $\sigma_x$. The spin-dependent beam splitter $\pi$-pulse, $\operatorname{SBS}(\pi)$, then maps the joint parity information onto the spin state. Finally, the expectation value of the joint parity operator is extracted from the spin-state population, and the result is rescaled by a factor of $4/\pi^2$ to obtain the Wigner function value. The diagram was generated using Quantikz Alastair2018.
• Figure 2: Circuit diagram representing the multimode joint-parity measurement scheme. $U_\psi$ prepares a $2N$-mode bosonic state, and each spin-dependent beam splitter $\pi$-pulse, $\operatorname{SBS}_j(\pi)$, encodes the joint parity of the mode pair $(2j-1, 2j)$ onto the spin.
• Figure 3: Schematic diagram of the radial RF and ground electrodes along with the principal axes of the radial motional modes $1$ and $2$. Each opposing electrode pair is shorted. A DC offset is applied to the RF electrode pair aligned with the mode $1$, while the other pair is grounded. The vector $\Delta k$ indicates the momentum kick imparted by the Raman transition.
• Figure 4: Oscillation of spin population under spin-dependent beam splitter interaction for Fock states $\ket{1}\ket{0}$ and $\ket{0}\ket{2}$. Scatter points show experimental data, while the solid and dashed curves represent simulations including motional heating. Error bars reflect quantum projection noise from 300 repetitions per data point. Dotted curves illustrate the noiseless simulation. The black dash-dotted line indicates $t_\pi$, at which the joint parity is mapped onto the spin population.
• Figure 5: Wigner distributions $W(\beta_1, \beta_2)$ of two-mode Fock states. (a) Fock state $\ket{1}\ket{1}$; (b) Fock state $\ket{2}\ket{1}$. The "Experiment" and "Simulation" columns depict the experimentally measured and numerically simulated Wigner functions, respectively. See Appendix \ref{['sect:numerical-simulation']} for the noise-aware simulation. The experiments are repeated by 300 shots per data point.